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Vanishing black dots

One of my earliest visual illusion experiments. Fix your gaze on the blinking dot at the center of the moving pattern. After a short while, one or more of the black dots may vanish and reappear on their own—sometimes singly, sometimes in pairs, occasionally all three.

Sustained fixation tends to trigger the disappearance; even a small shift of gaze brings them back.

The animated pattern is based on research by Yoram Bonneh, published in Nature under the phenomenon known as motion-induced blindness or MIB.

☞ More of my earliest optical illusions—naive by design, deceptive by nature.

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The Golden Triangle of Infinite Subdivision

This figure illustrates a remarkable self-similar properties of the golden ratio φ.

In the diagram, each successive level of the equilateral triangle is scaled by the factor φ⁻¹ ≈ 0.618. The linear dimensions of the triangles therefore follow the geometric progression:

φ⁻¹ , φ⁻² , φ⁻³ , φ⁻⁴ , …

Each level introduces smaller equilateral triangles arranged in a triangular lattice. The pattern approaches the base through an infinite cascade of similar shapes, producing a triangular fractal.

The infinite sum of these powers converges to the value of φ itself:

φ = Σ φ⁻ⁿ (from n = 1 to ∞)

This elegant equality explains why the golden ratio is closely tied to self-similarity and continued fractions. The figure offers a geometric illustration of the identity φ = 1 + 1/φ, expressed as an infinite cascade of progressively smaller triangles.

This illustration is taken from my forthcoming book on special constants — φ, π, e and their fascinating kin — presented in the style of old mathematical engravings.

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Miquel’s Pentagram Theorem

Consider a convex pentagon and extend its sides to form a pentagram. This produces five triangles outside the original pentagon. Construct the circumcircle of each triangle.

Each pair of adjacent circles intersects at two points: one is a vertex of the pentagon, and the other is a distinct second point.

Miquel’s Pentagram Theorem states that these five second points are concyclic; in other words, they all lie on a single common circle.

Miquel Circles
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When an Equilateral Triangle Goes Golden

Even an equilateral triangle can’t escape the golden ratio.
Divide one side so a / b = φ⁻¹ (≈ 0.618), and you get this elegant self-similar triangular spiral shrinking inward — same proportion at every step.

This illustration is taken from my forthcoming book on special constants — φ, π, e and their fascinating kin — presented in the style of old mathematical engravings.


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Natural Curve

A single blade of grass, a perfect mathematical curve.

Photo by Alexandre Manuel, landscape, Vietnam, 2017.

This delicate stem hanging over the water is a beautiful real-world example of a “catenary curve” — the same natural shape formed by a hanging chain, cable, or power line under gravity.

Mathematically described as:

y = a · cosh(x/a) + c

where a = T₀ / (μ · g)

and:

T₀ is the horizontal tension at the lowest point,

μ is the linear mass density,

g is the acceleration due to gravity.

In nature, flexible plants and vines spontaneously adopt this elegant form when influenced only by their own weight and gravity. The small leaves here act as gentle point loads, creating a soft, organic variation of the ideal catenary.

From suspension bridges to power lines to hanging droplets — the same beautiful mathematics appears everywhere.

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Iconic Mesoamerican Calendar

Aztec Stone of the Sun at the Museo Nacional de Antropología, Mexico.

“La Piedra del Sol“, often miscalled the Aztec Calendar, is one of the most iconic works of Aztec art. Carved from a single basalt block (not specifically olivine basalt, which is a common misattribution), this disk encapsulates the cosmological and chronological concepts of ancient Mexico.

Mathematically and astronomically, the stone encodes the Mexican dual calendrical system. Around the center are the 20 day‑signs of the sacred “tōnalpōhualli” (260‑day ritual cycle), while broader bands reference the 365‑day solar year (“xiuhpōhualli”). The interlocking 260‑ and 365‑day cycles form a 52‑year calendar round, a period when both counts realign—demonstrating sophisticated cyclic timekeeping.

From a scientific perspective, this pairing of cycles reflects a fundamental rhythm of Earth’s orbit and ritual time. The calendar round of 18,980 days (~52 years) arises from the arithmetic relation:

365×52 = 260×73,

showing how two distinct cycles synchronize over long spans—akin to the concept of a least common multiple in mathematics.

☞ Further reading.

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The Venn That Adds Up

The numbers 1–7 are placed in the regions of this diagram so that each circular set has the same total—19. The result is a “magic Venn diagram,” a type of arrangement explored by mathematician David G. Robinson of the University of West Georgia.

➡️ More number facts.

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TETRABLERONE – UNOFFICIAL RECEIPT

Born in Switzerland, I grew up eating countless Toblerones—those perfect equilateral prisms of chocolate.
Here’s a receipt for building a twelve-prism structure that will make even math teachers pause.

Ingredients

  • 12 Toblerone bars, in original wrappers (three of each flavor)
  • 8 rubber bands

Instructions
No melting. No unwrapping. Just follow the picture. The bars interlock to form the structure—precision and confidence recommended.

12 prism Toblerone

Result
One geometric confection. Edible? Technically. Stable? Debatable. Memorable? Absolutely.

Fun Geometry Note
Replace the Toblerones with simple sticks, and you’ll see it clearly: a three-dimensional cross of twelve sticks, three along each long diagonal of an imaginary cube.

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Squares with digits 1, 4, 9

At Carnegie Mellon University, Jacobson and Applegate uncovered a remarkable number:
419994999149149944149149944191494441.

What makes it remarkable is that every digit is a perfect square—1, 4, or 9—and the number itself is also a non-zero perfect square: 648070211589107021².

With roughly 4.199 × 10³⁵, it is believed to be the largest known square with this property. The researchers confirmed that no other number smaller than 10⁴² shares this exact pattern.

The preceding squares in this series, composed solely of the digits 1, 4, and 9, are:
1, 4, 9, 49, 144, 441, 1444, 11449, 44944, 991494144, 4914991449, 149991994944, 9141411499911441, 199499144494999441, 9914419419914449449, 444411911999914911441.

☞ More number facts.